Hypercytokinemia, or cytokine storm, is one of the severe complications of viral and bacterial infections, involving the release of abnormal amounts of cytokines, resulting in a massive inflammatory response. Cytokine storm is associated with COVID-19 and sepsis high mortality rate by developing epithelial dysfunction and coagulopathy, leading to thromboembolism and multiple organ dysfunction syndrome. Anticoagulant therapy is an important tactic to prevent thrombosis in sepsis and COVID-19, but recent data show the incompatibility of modern direct oral anticoagulants and antiviral agents. It seems relevant to develop dual-action drugs with antiviral and anticoagulant properties. At the same time, it was shown that azolo[1,5-a]pyrimidines are heterocycles with a broad spectrum of antiviral activity. We have synthesized a new family of azolo[1,5-a]pyrimidines and their condensed polycyclic analogs by cyclocondensation reactions and direct CH-functionalization and studied their anticoagulant properties. Five compounds among 1,2,4-triazolo[1,5-a]pyrimidin-7-ones and 5-alkyl-1,3,4-thiadiazolo[3,2-a]purin-8-ones demonstrated higher anticoagulant activity than the reference drug, dabigatran etexilate. Antithrombin activity of most active compounds was confirmed using lipopolysaccharide (LPS)-treated blood to mimic the conditions of cytokine release syndrome. The studied compounds affected only the thrombin time value, reliably increasing it 6.5–15.2 times as compared to LPS-treated blood.